Topology is weird: The Ham Sandwich Theorem

Keep exploring at ► https://brilliant.org/TreforBazett. Get started for free, and hurry—the first 200 people get 20% off an annual premium subscription.

Today we talk about my favourite math theorem: the Ham Sandwich theorem. Consider a sandwich with three components. Then the theorem claims there exists some plane in three dimensions, where if I slice along that plane then I cut all three components exactly in half. We will see repeated use of the Intermediate Value Theorem in the simpler cases (which we call the pancake theorem), and then the more powerful Borsuk-Ulam theorem to prove the main result.

0:00 The Ham Sandwich Theorem
0:56 One Pancake and the Intermediate Value Theorem
2:12 The two pancakes theorem
3:40 Borsuk-Ulam Theorem
6:07 Proving the Ham Sandwich Theorem
10:45 Higher dimensions
11:24 Brilliant.org/TreforBazett

My previous video proving Sperner's Lemma and Brouwer's Fixed Point theorem (which can be shown are equivalent to Borsuk-Ulam):    • A beautiful combinatorical proof of the Br...  

I also have a previous video on Borsuk-Ulam and some other cool corollaries. It was actually my 10k subs video!:    • My Favorite Theorem: The Borsuk-Ulam Theorem  

COURSE PLAYLISTS:
►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Logic, P...  
►LINEAR ALGEBRA:    • Linear Algebra (Full Course)  
►CALCULUS I:    • Calculus I (Limits, Derivative, Integrals)...  
► CALCULUS II:    • Calculus II (Integration Methods, Series, ...  
►MULTIVARIABLE CALCULUS (Calc III):    • Calculus III: Multivariable Calculus (Vect...  
►VECTOR CALCULUS (Calc IV)    • Calculus IV: Vector Calculus (Line Integra...  
►DIFFERENTIAL EQUATIONS:    • Ordinary Differential Equations (ODEs)  
►LAPLACE TRANSFORM:    • Laplace Transforms and Solving ODEs  
►GAME THEORY:    • Game Theory  

OTHER PLAYLISTS:
► Learning Math Series
   • 5 Tips To Make Math Practice Problems Actu...  
►Cool Math Series:
   • Cool Math Series  

BECOME A MEMBER:
►Join:    / @drtrefor  

MATH BOOKS I LOVE (affilliate link):
► https://www.amazon.com/shop/treforbazett

SOCIALS:
►Twitter (math based):   / treforbazett  
►Instagram (photography based):   / treforphotography